Assignment P. 298-301: 1-6, 8, 10, 12-19 some, 20, 21, 24, 29, 30, 36, 37, 47, 48 Challenge Problems

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Transcript of Assignment P. 298-301: 1-6, 8, 10, 12-19 some, 20, 21, 24, 29, 30, 36, 37, 47, 48 Challenge Problems

5.1 Midsegment Theorem 5.2 Use Perpendicular Bisectors

AssignmentP. 298-301: 1-6, 8, 10, 12-19 some, 20, 21, 24, 29, 30, 36, 37, 47, 48 Challenge Problems

Warm-UpA midsegment of a triangle is a segment that connects the midpoints of two sides of the triangle. Every triangle has 3 midsegments.

Warm-UpOn a piece of patty paper, draw a large acute ABC.Find the midpoints of each side by putting two vertices on top of each other and pinching the midpoint.

Warm-UpLabel the midpoints M, N, and P. Draw the three midsegments of your triangle by connecting the midpoints of each side.

Warm-UpUse another piece of patty paper to trace off AMP.

Warm-UpCompare all the small triangles. What do you notice about the length of a midsegment and the opposite side of the triangle? What kind of lines do they appear to be?

Warm-UpCompare all the small triangles. What do you notice about the length of a midsegment and the opposite side of the triangle? What kind of lines do they appear to be?

Warm-UpCompare all the small triangles. What do you notice about the length of a midsegment and the opposite side of the triangle? What kind of lines do they appear to be?

5.1 Midsegment Theorem and Coordinate ProofObjectives:To discover and use the Midsegment TheoremTo write a coordinate proof95.1 Midsegment Theorem and Coordinate ProofMidsegmentA midsegment of a triangle is a segment that connects the midpoints of two sides of the triangle. Every triangle has 3 midsegments.

MidsegmentA midsegment of a triangle is a segment that connects the midpoints of two sides of the triangle. Every triangle has 3 midsegments.

Example 1Graph ACE with coordinates A(-1, -1), C(3, 5), and E(7, -5). Graph the midsegment MS that connects the midpoints of AC and CE.

Example 1Now find the slope and length of MS and AE. What do you notice about the midsegment and the third side of the triangle?

Midsegment Theorem

The segment connecting the midpoints of two sides of a triangle is parallel to the third side and is half as long as that side.Example 2The diagram shows an illustration of a roof truss, where UV and VW are midsegments of RST. Find UV and RS.

Example 3

1.2.Deep, Penetrating QuestionsHow many examples did we look at to come up with our Theorem?

Is that enough?

How could we prove this theorem?

Where could we prove this theorem?Coordinate ProofCoordinate proofs are easy. You just have to conveniently place your geometric figure in the coordinate plane and use variables to represent each vertex.These variables, of course, can represent any and all cases.When the shape is in the coordinate plane, its just a simple matter of using formulas for distance, slope, midpoints, etc.Example 4 Place a rectangle in the coordinate plane in such a way that it is convenient for finding side lengths. Assign variables for the coordinates of each vertex.

Example 4 Convenient placement usually involves using the origin as a vertex and lining up one or more sides of the shape on the x- or y-axis.

Example 5 Place a triangle in the coordinate plane in such a way that it is convenient for finding side lengths. Assign variables for the coordinates of each vertex.

Example 6Place the figure in the coordinate plane in a convenient way. Assign coordinates to each vertex.Right triangle: leg lengths are 5 units and 3 unitsIsosceles Right triangle: leg length is 10 unitsExample 7A square has vertices (0, 0), (m, 0), and (0, m). Find the fourth vertex.

Example 8Find the missing coordinates. The show that the statement is true.

Example 9Write a coordinate proof for the Midsegment Theorem.

Given: MS is a midsegment of OWL

Prove: MS || OL and MS = OLExample 10Explain why the choice of variables below might be slightly more convenient.Given: MS is a midsegment of OWL

Prove: MS || OL and MS = OL

AssignmentP. 298-301: 1-6, 8, 10, 12-19 some, 20, 21, 24, 29, 30, 36, 37, 47, 48 Challenge Problems